On the maximum number of minimal codewords

TL;DR

This paper investigates the maximum number of minimal codewords in binary linear codes, providing new bounds, exact values for specific cases, and a formula for certain code parameters, advancing understanding in coding theory and cryptography.

Contribution

It offers improved bounds, exact counts for specific code lengths and dimensions, and a new formula for minimal codewords in certain linear codes.

Findings

Derived improved lower and upper bounds on the maximum number of minimal codewords.

Determined exact values for codes with length = dimension + 2.

Provided a formula for counting minimal codewords in codes with length = dimension + 3.

Abstract

Minimal codewords have applications in decoding linear codes and in cryptography. We study the maximum number of minimal codewords in binary linear codes of a given length and dimension. Improved lower and upper bounds on the maximum number are presented. We determine the exact values for the case of linear codes of dimension $k$ and length $k+2$ and for small values of the length and dimension. We also give a formula for the number of minimal codewords of linear codes of dimension $k$ and length $k+3$.